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\author{五六七}
\title{常微分方程第5周作业（5.1）}
%\date{2025年10月9日}

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%\abstract{第6章第1节：例1、例子2、习题1(1)、2(1)。}

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%自治的微分方程的降阶法
\item  将自治的二阶微分方程 $y''+yy'=1$ 化为一阶的微分方程。

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\item  将下述自治的三阶微分方程化为二阶的微分方程：
$$F\left( y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \frac{d^3y}{dx^3}\right) = 0. $$

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\item  求解二阶自治微分方程 $\frac{d^2x}{dt^2}=f(x)$ 并画出相图。

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\item  求解微分方程 $\frac{d^2x}{dt^2} = -x$. 记 $v=\frac{dx}{dt}$, 画出 $(x,v)$ 的相图。

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\item  求解微分方程 $\frac{d^2x}{dt^2} = x$. 记 $v=\frac{dx}{dt}$, 画出 $(x,v)$ 的相图。

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\item  将三阶微分方程 $y'''=F(x,y,y',y'')$ 写成三阶标准微分方程组。

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\item  求下述微分方程组（其中的求导均是相对于自变量 $x$ 而言）的阶数，并将其写成标准微分方程组：
\begin{eqnarray*}
\left\{\begin{array}{lcl}
u\,'' &=& F(x,u,u\,',v,w,w\,',w\,''), \\
v\,' &=& G(x,u,u\,',v,w,w\,',w\,''), \\
w\,''' &=& H(x,u,u\,',v,w,w\,',w\,'').
\end{array}\right.
\end{eqnarray*}

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\item  例子5.1. 单摆的微分方程。
设摆线长为 $\ell$, 设小球的质量为 $m$. 设摆线与垂线的有方向的夹角为 $x$. 
导出单摆方程、求解、画出单摆运动的相图。

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